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Mathematics for Physics

34 Gauge

A choice of potentials for the electromagnetic field that is used to fix unphysical freedom while leaving the physical fields unchanged.

Note: Also called a gauge choice. Also called working in a gauge.

definition [d] (Gauge Freedom = Gauge Transformation) From Griffiths: Eqs. for the potentials do not uniquely define \(V\) and \(\mathbf{A}\); we are free to impose extra conditions on \(V\) and \(\mathbf{A}\), as long as nothing happens to \(\mathbf{E}\) and \(\mathbf{B}\). For any scalar function \(\lambda(\mathbf{r}, t)\), we can add \(\nabla\lambda\) to \(\mathbf{A}\), provided we simultaneously subtract \(\partial\lambda/\partial t\) from \(V\). This will not affect the physical quantities \(\mathbf{E}\) and \(\mathbf{B}\). Such changes in \(V\) and \(\mathbf{A}\) are called gauge transformations:

  • \(\mathbf{A}' = \mathbf{A} + \nabla\lambda\) ,
  • \(V' = V - \dfrac{\partial\lambda}{\partial t}\) .

where

  • \(V\) is the electric scalar potential.
  • \(\mathbf{A}\) is the magnetic vector potential.
  • \(\lambda\) is an arbitrary scalar function of position and time.
  • \(\mathbf{E}\) and \(\mathbf{B}\) are the physical electric and magnetic fields.

definition [d] (Gauge Transformation) From Frankel: a local change of basis, such as

  • \(e_{V} = e_{U}\, c_{UV}\) ,

is called in physics a gauge transformation. Gauge transformations are simply changes of frames in the fibers of the bundle.

where

  • \(e_{U}\) and \(e_{V}\) are local frames on overlapping regions.
  • \(c_{UV}\) is the transition function relating those frames.

34.1 Elementary Example

34.1.1 Simple

In magnetostatics, the Coulomb gauge is the choice \(\nabla\cdot\mathbf{A} = 0\).

\[ \mathbf{A}' = \mathbf{A} + \nabla\lambda,\quad \nabla\cdot\mathbf{A}' = 0 \]

where

  • \(\lambda\) is chosen so that the new vector potential is divergenceless.

34.1.2 General

A full electrodynamic gauge change shifts both potentials together.

\[ \mathbf{A}' = \mathbf{A} + \nabla\lambda \]

\[ V' = V - \dfrac{\partial\lambda}{\partial t} \]

\[ \mathbf{E}' = \mathbf{E},\quad \mathbf{B}' = \mathbf{B} \]

where

  • the physical fields are unchanged by the gauge choice.

34.2 References

  1. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — gauge freedom and gauge transformations of \(V\) and \(\mathbf{A}\).
  2. Frankel, T. The Geometry of Physics: An Introduction. Cambridge University Press, 2012. — gauge transformation as a local change of fiber frame.
  3. Hubbard, J. H., & Hubbard, B. B. Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. Matrix Editions, 2015. — working in a different gauge as a change of bundle coordinates.